ON THE INTERSECTION THEORY OF QUOT SCHEMES AND MODULI OF BUNDLES WITH SECTIONS

ON THE INTERSECTION THEORY OF QUOT SCHEMES AND MODULI OF BUNDLES WITH SECTIONS ALINA MARIAN Abstract. We consider a class of tautological top intersection products on the moduli space of stable pairs consisting of vector bundles together with N sections on a smooth complex curve C. We show that when N is large, these intersection numbers can be viewed as arising on the Grothendieck Quot scheme of coherent sheaf quotients of the rank N trivial sheaf on C. The result has applications to the computation of the intersection theory of the moduli space of semistable bundles on C. 1. Introduction We study an intersection-theoretic relationship between two moduli spaces associated with bundles on a curve. We fix throughout a smooth complex curve C of genus g. One of the two moduli spaces is the Grothendieck Quot scheme Quot d (O N, r, C) parametrizing rank N r degree d coherent sheaf quotients of O N on C. The other is a parameter space, subsequently denoted B d (N, r, C), for stable pairs consisting of a rank r degree d vector bundle on C together with N sections. Several moduli spaces of pairs of vector bundles and sections can be constructed gauge-theoretically or by GIT, by varying the notion of stability of a pair [BDW], [Th]. B d (N, r, C) is one of these, and it is the one most closely related to the moduli space M(r, d) of semistable rank r and degree d bundles on C. Indeed, a point in B d (N, r, C) consists of a semistable degree d bundle E and N sections s 1,..., s N such that any proper subbundle of E whose space of sections contains s 1,..., s N is not slope-destabilizing. We will assume throughout that N 2r + 1, and d is as large as convenient, in particular that d > 2r(g 1). In this case, B d (N, r, C) is smooth projective [BDW] and there is a surjective morphism forgetting the sections, π : B d (N, r, C) M(r, d). In turn, for d large relative to N, r, g, Quot d (O N, r, C) is irreducible, generically smooth, of the expected dimension [BDW]. The closed points of Quot d (O N, r, C) are exact sequences 0 E O N F 0 on C with F coherent of degree d and rank N r. Thus Quot d (O N, r, C) compactifies the scheme Mor d (C, G(r, N)) of degree d morphisms from C to the Grassmannian G(r, N), which is the locus of exact sequences on C with F locally free of degree d and rank N r. The two spaces B d (N, r, C) and Quot d (O N, r, C) are birational, agreeing on the open subscheme U of Quot d (O N, r, C) consisting of sequences 0 E O N F 0 with E semistable: this sequence in Quot d (O N, r, C) corresponds to the point O N E 1

2 ALINA MARIAN in B d (N, r, C). Both Quot d (O N, r, C) and B d (N, r, C) are fine moduli spaces carrying universal structures. We will denote by 0 E O N F 0 on Quot d (O N, r, C) C the universal exact sequence associated with Quot d (O N, r, C) and by O N V on B d (N, r, C) C. the universal bundle with N sections corresponding to B d (N, r, C). Here E and V are locally free, and O N V coincides with O N E on the common open subscheme U of B d (N, r, C) and Quot d (O N, r, C). We further fix a symplectic basis {1, δ j, 1 j 2g, ω} for the cohomology H (C) of the curve C and let (1) c i (E ) = a i 1 + (2) c i (V) = ā i 1 + 2g j=1 b j i δ j + f i ω, 1 i r, and 2g j=1 bj i δ j + f i ω, 1 i r, be the Künneth decompositions of the Chern classes of E and V with respect to this basis for H (C). Note that a i H 2i (Quot d (O N, r, C)), b j i H2i 1 (Quot d (O N, r, C)), f i H 2i 2 (Quot d (O N, r, C)), and ā i, b j i, f i belong to the analogous cohomology groups of B d (N, r, C). The goal of this paper is to present the following observation. Theorem 1. Let r 2 and N 2r + 1. Assume moreover that C has genus at least 2 and that d is large enough so that both Quot d (O N, r, C) and B d (N, r, C) are irreducible of the expected dimension e = Nd r(n r)(g 1). Let s and t be nonnegative integers such that t N r and s r + t = e. Let P (a, f, b) be a polynomial of degree t in the a, f and (algebraic combinations) of the b classes. Then (3) ā s rp (ā, f, b) = a s rp (a, f, b). B d (N,r,C) Quot d (O N,r,C) Note that the theorem by no means accomplishes a complete comparison of the intersection theories of B d (N, r, C) and of Quot d (O N, r, C). Its scope is in fact modest: it asserts that the intersection of any algebraic polynomial P (a, f, b) in the tautological classes paired with a complementary self-intersection a s r can be evaluated to the same result on Quot d (O N, r, C) and on B d (N, r, C) only asymptotically as N is large relative to the degree of P. A more general statement is false due to the different nature of the nonoverlapping loci of Quot d (O N, r, C) and B d (N, r, C).

INTERSECTION THEORY OF QUOT SCHEMES AND MODULI OF BUNDLES WITH SECTIONS 3 The proof of the theorem, which will occupy the next section, relies on an explicit realization of the tautological classes on B d (N, r, C) and Quot d (O N, r, C) as degeneracy loci (or pushforwards of degeneracy loci) of morphisms stemming from the universal O N V on B d (N, r, C) C and O N E on Quot d (O N, r, C) C. This approach was used in [B] to study the intersection theory of the tautological a classes on Quot d (O N, r, C). Bertram proved that generic degeneracy-loci representatives of the a classes in a top-degree polynomial P (a 1,..., a r ) intersect properly along Mor d (C, G(r, N)) and avoid the boundary Quot d (O N, r, C)\Mor d (C, G(r, N)). Thus the intersection P (a 1,..., a r ) Quot d (O N, r, C) has enumerative meaning, and expresses a count of maps from C to G(r, N) satisfying incidence properties. Similarly, in the situation described by Theorem 1, we will show that generic degeneracyloci representatives of the a, f, b classes in the product a s rp (a, f, b) on the one hand intersect properly along the common open subscheme U of Quot d (O N, r, C) and B d (N, r, C) and on the other hand manage to avoid the nonoverlapping loci B d (N, r, C) \ U and Quot d (O N, r, C) \ U. The argument uses heavily the methods and results of [B]. The theorem was sought, and is indeed effective, in the attempt to calculate the intersection theory of the moduli space M(r, d) of semistable bundles on C, by transferring intersections to the Quot scheme Quot d (O N, r, C) and computing them there. This idea was inspired by [W]. The intersection-theoretic relationship between M(r, d) and Quot d (O N, r, C) is simplest when the rank r and the degree d are coprime. In this case, M(r, d) is a smooth projective variety and a fine moduli space with a universal bundle V M(r, d) C. Furthermore, B d (N, r, C) is a projective bundle over M(r, d), π : B d (N, r, C) = P(η V N ) M(r, d), where η : M(r, d) C M(r, d) is the projection. Intersections on M(r, d) can therefore be easily lifted to B d (N, r, C). To be precise, let c i (V ) = ã i 1 + 2g j=1 bj i δ j + f i ω, 1 i r, be the Künneth decomposition of the Chern classes of V, and let ζ = c 1 (O(1)) on B d (N, r, C) = P(η V N ). It is easy to see that V = π V O(1) on B d (N, r, C) C. We choose N so that the dimension N(d r(g 1)) 1 of the fibers of π is a multiple of r, N(d r(g 1)) 1 = rs. Then if P (ã, b, f) is a top-degree polynomial on M(r, d) invariant under twisting the universal V with line bundles from M(r, d), we have (4) M(r,d) P (ã, b, f) = B d (N,r,C) ζ dim fiber P (ā, b, f) = We therefore obtain the following consequence of Theorem 1. B d (N,r,C) ā s rp (ā, b, f). Corollary 1. For r and d coprime, top intersections on M(r, d) can be viewed as intersections on Quot d (O N, r, C) in the regime when N is large relative to the dimension

4 ALINA MARIAN of M(r, d), and d is large enough so that Quot d (O N, r, C) is irreducible. Specifically, for N(d r(g 1)) 1 divisible by r and N r dim M(r, d), (5) P (ã, b, f) = a s rp (a, b, f). M(r,d) Quot d (O N,r,C) The calculation of the tautological intersection theory on the Quot scheme is manageable via equivariant localization with respect to a natural C action [MO1]. Theorem 1 together with the results of [MO1] thus constitute an effective method for evaluating the top pairings of the cohomology generators of M(r, d), and provide an alternative to the important work [JK], [L] on the intersection theory of M(r, d). In the rank 2, odd degree case, complete calculations were carried out in [MO2]. A derivation of the volume of M(r, d) as well as of other intersection numbers for arbitrary r is pursued in [MO3]. Acknowledgements. The research for this paper was carried out during the author s graduate work at Harvard and she would like to thank her advisor, Shing-Tung Yau, for his guidance and support. Conversations with Dragos Oprea were as usual pleasant and instructive. 2. Tautological intersections on Quot d (O N, r, C) and B d (N, r, C) Following [B] closely, we first summarize the main facts concerning the boundary of the compactification by Quot d (O N, r, C) of the space Mor d (C, G(r, N)) of maps from C to the Grassmannian G(r, N). We then analyze the relevant intersections on Quot d (O N, r, C) and B d (N, r, C), with conclusions gathered in Proposition 1 and Proposition 2 respectively. Theorem 1 follows from the two propositions. 2.1. The Quot scheme compactification of the space of maps to the Grassmannian G(r, N). Quot d (O N, r, C) parametrizes exact sequences 0 E O N F 0 with F coherent of degree d and rank N r. Two quotients O N F 1 0 and O N F 2 0 are equivalent if their kernels inside O N coincide. Now let 0 S O N Q 0 denote the tautological sequence on G(r, N). By pullback of the tautological sequence to C, a degree d map f : C G(r, N) gives a short exact sequence 0 E O N F 0 of vector bundles on C and conversely. Thus Mor d (C, G(r, N)) is contained in Quot d (O N, r, C) as the locus of locally free quotients of O N on C. There are natural evaluation morphisms (6) ev : Mor d (C, G(r, N)) C G(r, N), (f, p) f(p), and for p C, (7) ev p : Mor d (C, G(r, N)) G(r, N), f f(p).

INTERSECTION THEORY OF QUOT SCHEMES AND MODULI OF BUNDLES WITH SECTIONS 5 The Zariski tangent space at a closed point f Mor d (C, G(r, N)) is H 0 (C, f T G(r, N)). The expected dimension, which we subsequently denote by e, is therefore e = χ(c, f T G(r, N)) = Nd r(n r)(g 1), by the Riemann-Roch formula on C. In [BDW], Bertram, Daskalopoulos and Wentworth established For sufficiently large degree d relative to r, N, and the genus g of C, Mor d (C, G(r, N)) is an irreducible quasiprojective scheme of dimension as expected, e. In this regime, the compactification Quot d (O N, r, C) is an irreducible generically smooth projective scheme. We assume throughout this paper that the degree is very large compared to r, N and g. What remains in Quot d (O N, r, C) once we take out Mor d (C, G(r, N)) is the locus of quotients of O N which are not locally free. Since a torsion-free coherent sheaf on a smooth curve is locally free, these nonvector bundle quotients are locally free modulo a torsion subsheaf which they necessarily contain. That is to say, for each such O N F 0 there is an exact sequence 0 T F F/T 0 with F/T locally free of degree at least 0, and T supported at finitely many points. The degree of T can therefore be at most d. Let B m be the locus in Quot d (O N, r, C) of quotients whose torsion subsheaf T has degree m. Quot d (O N, r, C) = Mor d (C, G(r, N)) d m=1 B m, with the B m s playing the role of boundary strata. d m=1 B m. We will denote by B the union We analyze B more carefully. If F is a quotient with torsion T of degree m, F/T is a vector bundle of rank N r and degree d m. By associating to O N F 0 the quotient O N F/T 0, we get a morphism (8) P : B m Mor d m (C, G(r, N)), {O N F } {O N F/T }. Let us consider the inverse image under P of an element 0 E O N F 0 in Mor d m (C, G(r, N)). For each sheaf morphism E T, with T torsion of degree m, there is a sheaf F constructed as the pushout of the maps E O N and E T in the category of coherent sheaves on C. Thus the following diagram of sheaves can be completed, if we start with the data of the first row and of the first column map: 0 E O N F 0 0 T F F 0 Moreover the map O N F is surjective if E T is. If we denote its kernel by E we have

6 ALINA MARIAN E = E 0 E O N F 0, 0 T F F 0 with the first two columns also being short exact sequences. Finally assume that two quotients O N F 1 and O N F 2 induce quotients O N F 1 /T 1 and O N F 2 /T 2 which have the same kernel E inside O N. It is then immediate that O N F 1 and O N F 2 represent the same element of Quot d (O N, r, C) if and only if E T 1 and E T 2 represent the same element of the Quot scheme of degree m, rank 0 quotients of E. This latter scheme, which we denote by Quot m (E, r, C), is a smooth projective variety of dimension rm. We conclude that the morphism is surjective with fiber over P : B m Mor d m (C, G(r, N)) 0 E O N F 0 isomorphic to Quot m (E, r, C). From this it follows easily that the codimension of the boundary stratum B m in Quot d (O N, r, C), for m small, is equal to (9) codim B m = (N r)m. The codimension of B m for m large (so that Mor d m (C, G(r, N)) is not irreducible) increases linearly with the degree d. Note also that by associating to the quotient O N F the double data of O N F/T and of the support of T, we get a morphism (10) P l : B m Sym m C Mor d m (C, G(r, N)), where Sym m C denotes the mth symmetric product of the curve C. 2.2. Intersections on the Quot scheme. The classes that we would like to intersect are the Künneth components (1) of c i (E ), 1 i r. Let be the projection, and note that η : Quot d (O N, r, C) C Quot d (O N, r, C) a i = c i (E p ), for p C, f i = η (c i (E )), 1 i r. Furthermore, combinations of b classes which are algebraic can be expressed in terms of a classes and of pushforwards under η of polynomials in the Chern classes of E. For instance, 2 g j=1 b j 1 bj+g 1 = 2da 1 η (c 2 1),

INTERSECTION THEORY OF QUOT SCHEMES AND MODULI OF BUNDLES WITH SECTIONS 7 g j=1 b j 1 bj+g 2 + g j=1 We will view an arbitrary homomorphism b j 2 bj+g 1 = a 1 η (c 2 ) + da 2 η (c 1 c 2 ). P (a, b, f) : A (Quot d (O N, r, C)) A deg P (Quot d (O N, r, C)) as an actual intersection operation of subschemes. For simplicity we will omit b classes from the discussion, since modulo a classes, they are pushforwards of polynomials in c i (E ), hence can be treated quite similarly to the f classes. Notation. For each i, 1 i r, we choose O r i+1, a trivial subbundle of O N on Quot d (O N, r, C) C. The universal exact sequence 0 E O N F 0 on Quot d (O N, r, C) C induces for each i, by dualizing and restricting, morphisms ϕ i : O r i+1 E. We let D i be the subscheme of Quot d (O N, r, C) C which is the noninjectivity locus of ϕ i. Similarly, by fixing p C, we denote by D i,p the subscheme of Quot d (O N, r, C) which is the degeneracy locus of the restriction ϕ i,p of ϕ i to Quot d (O N, r, C) p. We let D i A e i (D i ) and D i,p A e i (D i,p ) be the ith degeneracy classes of ϕ i and ϕ i,p respectively, satisfying D i = c i (E ) [Quot d (O N, r, C) C], D i,p = c i (E p ) [Quot d (O N, r, C)]. Denote by Di 0 and D0 i,p the restrictions of the degeneracy loci to Mor d(c, G(r, N)) C and Mor d (C, G(r, N)) respectively. Further, we let Y i be the degeneracy locus of the map O r i+1 S on G(r, N). It is well known that Y i represents the ith Chern class of S, and therefore has codimension i in G(r, N). Now Di 0 and Di,p 0 are precisely the pullbacks of Y i by the evaluation maps (6), (7), (11) D 0 i = ev 1 (Y i ), D 0 i,p = ev 1 p (Y i ). Thus Di 0 and Di,p 0 have codimension i in Mor d(c, G(r, N)) C and Mor d (C, G(r, N)) respectively, and Di,p 0 is the locus of degree d maps from C to G(r, N) which send the point p to Y i. Likewise, η(di 0) is the locus of maps whose images intersect Y i, and η(di 0) has codimension i 1 in Mor d (C, G(r, N)). This last statement follows because the fibers of the restricted η : Di 0 η(d0 i ) consist of finitely many points, outside a locus of high codimension in η(di 0 ). (This is the locus of maps f : C G(r, N) such that f(c) is contained in Y i, that is to say the morphism scheme Mor d (C, Y i ); its codimension in Mor d (C, G(r, N)) is proportional to the degree d of the maps, which is assumed very large.) In fact, for i 2, the fibers of the morphism η : Di 0 η(d0 i ) consist generically of exactly one point of the curve C. To see this, consider the double evaluation map ev 2 : Mor d (C, G(r, N)) C C G(r, N) G(r, N), (f, p, q) (f(p), f(q)). Let η 2 be the projection Mor d (C, G(r, N)) C C Mor d (C, G(r, N)).

8 ALINA MARIAN The subvariety of the base η(di 0 ) over which η has fibers consisting of at least two points is η 2 ( ( ev 2) 1 (Yi Y i )) and has codimension 2i 2 in Mor d (C, G(r, N)), thus codimension i 1 in η(di 0) (here i 2.) The fibers of η : D0 i η(d0 i ) consist therefore generically of exactly one point, and η(di 0) has codimension i 1 in Mor d(c, G(r, N)). The codimension (9) of the boundary stratum B m is (N r)m and (N r)m > i, for 1 i r, 1 m d and for N 2r + 1. As the codimension of each irreducible component of the degeneracy locus D i is at most i on general grounds [F], Theorem 14.4, we conclude D i has codimension i in Quot d (O N, r, C) C, D i,p has codimension i in Quot d (O N, r, C), and η(d i ) has codimension i 1 in Quot d (O N, r, C). Now GL(N) acts on Quot d (O N, r, C) and on Mor d (C, G(r, N)) as the automorphism group of O N : for g GL(N), [π : O N F ] [π g : O N F ]. Translating degeneracy loci by the action of the group enables us to make intersections on Mor d (C, G(r, N)) proper. Indeed, for any subvariety Z of Mor d (C, G(r, N)) and a general g GL(N), the intersection Z gdi,p 0 is either empty or has codimension i in Z: by Kleiman s theorem, for a general g, the intersection ev p (Z) gy i is either empty or of the expected dimension in the Grassmannnian G(r, N), whence the intersection Z evp 1 (gy i ) also has the expected dimension. Similarly, for a general g GL(N) the intersection Z C ev 1 (gy i ) is either empty or has codimension i in Z C. Since for a general g the generic fiber of the projection η : Z C ev 1 (gy r i+1 ) Z ηev 1 (gy i ) is finite, the intersection Z ηev 1 (gy i ) = Z η(gdi 0 ) has codimension i 1 in Z. We let Mor d (C, G(r, N)) ss denote the smooth open subscheme of Mor d (C, G(r, N)) whose closed points are exact sequences 0 E O N F 0 with E semistable. The argument above shows that for a generic choice of trivial subbundle O r i+1, Di,p 0 intersects the unstable locus Mor d (C, G(r, N)) unstable = Mor d (C, G(r, N)) \ Mor d (C, G(r, N)) ss properly. For dimensional reasons, the restriction map A e i (D i,p ) A e i (D 0 i,p Mor d (C, G(r, N)) ss ) is therefore an isomorphism. We conclude that the degeneracy class D i,p is not only in the top Chow group of D i,p, but in fact D i,p = [D i,p ] since the equality holds for the restrictions to the smooth open subscheme Mor d (C, G(r, N)) ss of Mor d (C, G(r, N)). Similarly, D i = [D i ], and as η : D i η(d i ) is generically one-to-one, we have η [D i ] = [η(d i )].

INTERSECTION THEORY OF QUOT SCHEMES AND MODULI OF BUNDLES WITH SECTIONS 9 We collect these conclusions in the statement of the following lemma. Lemma 1. (1) For a generic choice of trivial subbundle O r i+1 of O N on Quot d (O N, r, C) C, [D i ] = c i (E ) [Quot d (O N, r, C) C], [D i,p ] = a i [Quot d (O N, r, C)] and [η(d i )] = f i [Quot d (O N, r, C)]. (2) For all instances of a i, f i and algebraic combinations of b j i classes in a monomial P (a, b, f) we can choose representative subschemes which intersect the unstable locus of Mor d (C, G(r, N)) and each other properly along Mor d (C, G(r, N)). We would now like to show that the intersection of generic degeneracy loci representatives of the tautological classes appearing in a monomial P (a, b, f)a s r, for e = sr + t, t = deg P and t N r avoids the boundary d m=1 B m of the Quot scheme. Lemma 1 ensures on the other hand that intersections occur with the expected codimension along Mor d (C, G(r, N)), even avoiding the unstable locus of Mor d (C, G(r, N)). We will thus be able to conclude that the intersection number P (a, b, f)a s r [Quot d (O N, r, C)] is the number of intersection points in Mor d (C, G(r, N)) ss (counted with multiplicities) of generic degeneracy loci representatives of all the classes in P (a, b, f)a s r. We therefore analyze the intersection of a degeneracy locus D i,p with a boundary stratum B m. The subtlety here is that although the B m s have large codimension in Quot d (O N, r, C), subschemes of the type D i,p and η(d i ) may not intersect them properly, therefore giving an overall excess intersection with the boundary. Suppose 0 E O N F 0 is a point in the intersection D i,p B m. This means that F has a torsion subsheaf T of degree m and the vector bundle map O r i+1 E is not injective at p. Recall the surjective morphism (10), P l : B m Sym m C Mor d m (C, G(r, N)), {O N F } {Supp T, O N F/T }, and its further composition P with the projection to Mor d m (C, G(r, N)), given by (8). Let 0 E m O N F m 0 be the universal sequence on Mor d m (C, G(r, N)) C and denote by D i,p (m) the degeneracy locus of Op r i+1 Ep m on Mor d m (C, G(r, N)). Let E be the kernel of the map O N F/T. If p / Supp T, then the map O r i+1 E is also not injective at p since it coincides with O r i+1 E outside the support of T. We conclude that (12) D i,p B m P 1 (D i,p (m)) P l 1 (p Mor d m (C, G(r, N))). We pick now s distinct points p k and s generic GL(N) elements g k, 1 k s. Each instance of a r in the product a s r is represented by a subscheme g k D r,pk, 1 k s. We want to show that (13) s k=1 g kd r,pk B m =.

10 ALINA MARIAN Indeed, if we let I index subsets of {1,..., s} of cardinality s m, then (12) implies that (14) s k=1 g kd r,pk B m I For general g k s, the subschemes g k D r,pk (m), 1 k s P 1 ( k I g k D r,pk (m)). intersect properly on Mor d m. The codimension of k I g k D r,pk (m) in Mor d m (C, G(r, N)) is therefore (15) (s m)r = e t mr Nd r(n r)(g 1) N r mr. The inequality above holds in the regime that we are interested in, when e rs = t N r. A dimension count for Mor d m (C, G(r, N)) will show now that k I g k D r,pk (m) is empty for all I with I = s m. There are two cases to be considered. Case 1. m is large enough so that Mor d m is not irreducible of the expected dimension. The dimension of Mor d m (C, G(r, N)) can be bounded in this case independently of the degree d. The codimension (15) of k I g k D r,pk (m) is however bounded below by an expression increasing linearly with d. For d sufficiently large therefore, the intersection k I g k D r,pk (m) is empty. Case 2. m is so that Mor d m is irreducible of the expected dimension N(d m) r(n r)(g 1). Comparing this dimension with the codimension (15) of k I g k D r,pk (m), we find N(d m) r(n r)(g 1) < Nd r(n r)(g 1) N mr, 1 m d, if N 2r+1. r In this case also therefore, the intersection s k=1 g kd r,pk B m is empty. Taking into account the analysis leading to Lemma 1, we have the following statement. Proposition 1. Generic degeneracy loci representatives of the classes appearing in the top intersection product P (a, f, b)a s r, deg P N r on Quot d (O N, r, C) do not intersect the boundary B = d m=1 B m of Quot d (O N, r, C) and intersect each other as well Mor d (C, G(r, N)) unstable properly along Mor d (C, G(r, N)). The intersection number P (a, f, b)a s r [Quot d (O N, r, C)] is therefore the number of points (counted with multiplicities) in the scheme-theoretic intersection of these degeneracy loci. All the intersection points lie in Mor d (C, G(r, N)) ss. 2.3. Intersections on the moduli space of stable pairs. For d > 2r(g 1), the moduli space B d (N, r, C) is a smooth projective variety which parametrizes morphisms ϕ : O N E on C with E being a semistable rank r and degree d bundle, and with the property that the subsheaf Im ϕ of E is not destabilizing. In other words, if Im ϕ has degree d and rank r, there is a strict inequality d (16) r < d r.

INTERSECTION THEORY OF QUOT SCHEMES AND MODULI OF BUNDLES WITH SECTIONS 11 This moduli space was described, together with other related moduli spaces of pairs of bundles and sections, in [BD],[BDW]. The case of bundles together with one section was studied in [Th]. B d (N, r, C) is a fine moduli space carrying a universal morphism Φ : O N V on B d (N, r, C) C. We let U be the open subscheme of B d (N, r, C) consisting of generically surjective (as maps of vector bundles) morphisms O N E. Crucially, U coincides with the semistable locus of Quot d (O N, r, C), i.e. with the open subscheme consisting of exact sequences 0 E O N F 0 with semistable E. The universal structures on Quot d (O N, r, C) and B d (N, r, C) agree on U: O N E on U C coincides with O N V on U C. The actions of GL(N) on Quot d (O N, r, C) and B d (N, r, C) also naturally coincide on U. In order to study some of the intersection theory of B d (N, r, C) in relation to that of the Quot scheme, it will be necessary to understand the boundary B d (N, r, C) \ U. Let B(r, d ), r < r be the locus of maps ϕ : O N E such that the locally free subsheaf Im ϕ of E has rank r < r and degree d. There are finitely many such loci since Moreover, 0 d < r r d and 1 r r 1. (17) B d (N, r, C) = U r <r,d B(r, d ). The B(r, d )s for r < r are strata in the flattening stratification of B d (N, r, C) for the sheaf Im Φ and the projection (slightly abusing notation) η : B d (N, r, C) C B d (N, r, C). We aim to investigate the codimension of B(r, d ) in B d (N, r, C). A quotient ϕ : O N E in B(r, d ) gives on the one hand a point Im ϕ O N in the Quot scheme Quot d (O N, r, C) of rank r, degree d subsheaves of O N. We denote this morphism by Ψ r,d, (18) Ψ r,d : B(r, d ) Quot d (O N, r, C), {ϕ : O N E} {Im ϕ O N }. On the other hand, ϕ : O N E determines a point Im ϕ E in the Quot scheme Quot d d (E, r, C) of rank r, degree d subsheaves of E. Conversely, a point in B(r, d ) is determined by a choice of semistable E in the moduli space M(r, d) of semistable rank r and degree d bundles on C, and a choice of points in Quot d d (E, r, C) and Quot d (O N, r, C) such that the subsheaf of E is isomorphic to the dual of the subsheaf of O N. Now [PR] provides an upper bound for the dimension of Quot d d (E, r, C) which is independent of the choice of E inside M(r, d), dim Quot d d (E, r, C) r (r r ) + dr d r. An upper bound for the dimension of B(r, d ) is then dim B(r, d ) dim Quot d (O N, r, C) + r (r r ) + dr d r + dim M(r, d).

12 ALINA MARIAN Recall that dim B d (N, r, C) = Nd r(n r)(g 1), and as established in [BDW], dim Quot d (O N, r, C) either has a degree-independent upper bound (for low d ) or else dim Quot d (O N, r, C) = Nd r (N r )(g 1). A lower bound for the codimension of B(r, d ) in B d (N, r, C) is thus codim B(r, d ) Nd Nd dr + d r + degree-independent terms. When d is very large, it follows that (19) codim B(r, d ) (N r)(d d ) > N r d. r For the last inequality we took account of the stability condition for morphisms ϕ : O N E expressed by (16). We conclude that codim B(r, d ) is very large when we assume d to be conveniently large. We seek now to establish an analogue of Lemma 1, that is to use the universal morphism Φ : O N V on B d (N, r, C) C in order to give degeneracy loci representatives of the tautological classes (2). For each i, 1 i r, we choose a subbundle O r i+1 of O N on B d (N, r, C) C. Denote by i and i,p the noninjectivity loci of O r i+1 V on B d (N, r, C) C and of Op r i+1 V p on B d (N, r, C) {p} respectively. i and D i agree of course on U C. We want to show as before Lemma 2. (1) For a generic choice of trivial subbundle O r i+1 of O N on B d (N, r, C) C, [ i ] = c i (V) [B d (N, r, C) C], [ i,p ] = ā i [B d (N, r, C)] and [η( i )] = f i [B d (N, r, C)]. (2) For all instances of ā i, fi and algebraic combinations of b j i classes in a monomial P (ā, b, f) we can choose representative subschemes which intersect each other properly along U. Proof. The proof is immediate. We need to show that i and i,p have codimension i in B d (N, r, C) C and B d (N, r, C) respectively, and that η : i η( i ) is generically one-to-one. But i U C has codimension i in U C since it coincides with D i there. In the same way i,p U has codimension i in U, and η : i η( i ) is generically one-to-one on U. On the other hand, the codimension of the boundary B d (N, r, C)\U = r <r,d B(r, d ) is very large in B d (N, r, C), hence the first part of the lemma follows. The second part of the lemma follows from the second part of Lemma 1. We aim to show that the intersection of generic degeneracy loci representatives of the tautological classes appearing in a monomial P (ā, b, f)ā s r, for e = sr + t, t = deg P and t N r avoids the boundary r <r,d B(r, d ) of B d (N, r, C). Lemma 1 ensures on the other hand that intersections occur with the expected codimension along U. We

INTERSECTION THEORY OF QUOT SCHEMES AND MODULI OF BUNDLES WITH SECTIONS 13 will thus be able to conclude that the intersection number P (ā, b, f)ā s r [B d (N, r, C)] is given as the sum of multiplicities of the finitely many intersection points in U of generic degeneracy loci representatives of all the classes in P (ā, b, f)ā s r. We analyze the intersection of a degeneracy locus r,p with a boundary stratum B(r, d ). r,p is the zero locus of a section O V p on B d (N, r, C). Let 0 E O N F 0 be the universal sequence on Quot d (O N, r, C) C. Let O be a subbundle of O N (corresponding under Ψ r,d to the one chosen before on B d(n, r, C)) and denote by p (r, d ) Quot d (O N, r, C) the zero locus of the restricted homomorphism O p E p on Quot d (O N, r, C)) {p}. Given ϕ : O N E, ϕ B(r, d ), let E = Im ϕ. Note that O E vanishes at p if and only if O E does. Hence r,p B(r,d ) = Ψ 1 r,d ( p (r, d )). We choose g k GL(N), p k C, 1 k s so that g k pk (r, d ) intersect properly on Quot d (O N, r, C). The intersection s k=1 g k pk (r, d ) is empty if its codimension exceeds the dimension of the ambient scheme Quot d (O N, r, C). We can assume that Quot d (O N, r, C) has the expected dimension Nd r (N r )(g 1). The intersection on the other hand has codimension sr = e t r Nd r(n r)(g 1) N r r > Nd r(n r )(g 1). r r The last inequality is equivalent to ( ) d (20) N r d r > (r r )(g 1) + N r 2, which holds since ( ) d N r d r N rr. We have just established the following Proposition 2. Generic degeneracy loci representatives of the classes appearing in the top intersection product P (ā, f, b)ā s r, deg P N r on B d (N, r, C) do not intersect the boundary B d (N, r, C) U = r <r,d B(r, d ) of B d (N, r, C) and intersect each other properly along U. The intersection number P (ā, f, b)ā s r [B d (N, r, C)] is therefore the number of points (counted with multiplicities) in the scheme-theoretic intersection of these degeneracy loci. All the intersection points lie in U. Since Quot d (O N, r, C) and B d (N, r, C), as well as their universal structures, agree on U, Propositions 1 and 2 imply Theorem 1.

14 ALINA MARIAN References [B] A. Bertram, Towards a Schubert Calculus for Maps from a Riemann Surface to a Grassmannian, Internat. J. Math 5 (1994), no 6, 811-825. [BDW] A. Bertram, G. Daskalopoulos and R. Wentworth, Gromov Invariants for Holomorphic Maps from Riemann Surfaces to Grassmannians, J. Amer. Math. Soc. 9 (1996), no 2, 529-571. [BD] S. B. Bradlow, G.D. Daskalopoulos, Moduli of stable pairs for holomorphic bundles over Riemann surfaces, Int. J. Math. 2 (1991), 477-513. [F] W. Fulton, Intersection theory, Springer Verlag 1998. [JK] L. Jeffrey, F. Kirwan, Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface, Ann. of Math. (2) 148 (1998), no. 1, 109-196. [L] K. Liu, Heat kernel and moduli space I, II, Math. Res. Lett 3 (1996), 743-762 and 4 (1997), 569-588. [M] A. Marian, Intersection theory on the moduli spaces of stable bundles via morphism spaces, Harvard University Thesis 2004. [MO1] A. Marian, D. Oprea, Virtual intersections on the Quot scheme and the Vafa-Intriligator formula, AG/0505685. [MO2] A. Marian, D. Oprea, On the intersection theory of the moduli space of rank two bundles, AG/0505422. [MO3] A. Marian, D. Oprea, in preparation. [PR] M. Popa, M. Roth, Stable maps and Quot schemes, Invent. Math. 152 (2003), no.3, 625-663. [Th] M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994), no. 2, 317 353. [W] E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, Geometry, topology, and physics, 357 422, Conf. Proc. Lecture Notes Geom. Topology, IV, Internat. Press, Cambridge, MA, 1995. Department of Mathematics Yale University, P.O. Box 208283, New Haven, CT, 06520-8283, USA E-mail address: alina.marian@yale.edu